稀疏逼近的MIP和集合覆盖方法
MIP and Set Covering approaches for Sparse Approximation
摘要：
稀疏近似问题要求找到一个解$x$，使得对于给定的范数$||\cdot||$，$||y - Hx|| < α$最小化支持$||x||_0 := \#\{j \ |\ x_j \neq 0 \}$的大小。我们提出了有效的不等式混合规划（MIP）制定这个问题，我们表明，这些家庭是足够的描述一组可行的支持。这导致了一个重新制定的问题，作为一个可编程（IP）模型，这反过来又代表了最小集覆盖制定，从而产生了许多家庭的有效的不等式，可用于加强模型。我们提出的算法来解决稀疏近似问题，包括分支切割的MIP，一个两阶段的算法，以解决集覆盖IP和启发式方法的基础上局部分支类型的约束。这些方法进行了比较，在计算实验的目标是测试其实际潜力。
英文摘要：
The Sparse Approximation problem asks to find a solution $x$ such that $||y - Hx|| < α$, for a given norm $||\cdot||$, minimizing the size of the support $||x||_0 := \#\{j \ |\ x_j \neq 0 \}$. We present valid inequalities for Mixed Integer Programming (MIP) formulations for this problem and we show that these families are sufficient to describe the set of feasible supports. This leads to a reformulation of the problem as an Integer Programming (IP) model which in turn represents a Minimum Set Covering formulation, thus yielding many families of valid inequalities which may be used to strengthen the models up. We propose algorithms to solve sparse approximation problems including a branch \& cut for the MIP, a two-stages algorithm to tackle the set covering IP and a heuristic approach based on Local Branching type constraints. These methods are compared in a computational experimentation with the goal of testing their practical potential.
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